3.6 Conclusion
4.2.1 Radiative coupling in series-connected ensembles
sion will occur. Ensembles with sub-cells that are vertically stacked but separated by a medium with a refractive index different than the cell material, as shown in panel (b), will send approximately half of radiative emission to the correct sub-cell.
Finally, ensembles where the cells are monolithically integrated, as in panel (c), can potentially result in most radiative emission being directed to the desired downstream sub-cell. As discussed in Chapter1, the radiative emission of a solar cell is affected by the cell’s optical environment. When the cell is placed on a substrate with a refractive index similar to the cell’s, light emitted internally in the direction of the substrate can pass directly out of the cell without reflection. In contrast, internally emitted light directed towards the top of the cell will not escape the cell unless it is inside the escape cone determined by the refractive index of air and the cell’s refractive index.
All the sub-cells in a monolithic stack typically have very similar refractive index val- ues, which conveniently allows high transmission of light through the device. Because sub-cells are substantially transparent to light emitted by sub-cells with lower band gap, in practice all sub-cells can be treated as having the same optical environment:
with air on top, and an index-matched substrate below. In this configuration, with an index of refraction of 1 for air and 3.6 for the semiconductor 93% of light escaping a cell will be transmitted to the sub-cell below it.
(a)
(b) (c)
Figure 4.4. Schematic of spectrum splitting sub-cell configurations and the potential for radiative coupling between sub-cells. The monolithic stack directs a high percentage of a sub-cell’s radiatively emitted photons to the sub-cell below, where they can be absorbed.
ensembles can be accounted for by the following procedure. First, consider the top two sub-cells of the ensemble. Each has a J-V relationship determined by the de- tailed balance equation. The series-connection constrains the sub-cells to operate at the same current density, resulting in equations 4.1 - 4.3 .
J1(V1) =Jabs1 −Jrad1up(V1)−Jrad1down(V1) (4.1)
J2(V2, V1) =Jabs2 −Jrad2up(V2)−Jrad2down(V2) +Jrad1down(V1) (4.2)
Jabs1 −Jrad1up(V1)−Jrad1down(V1) =Jabs2 −Jrad2up(V2)−Jrad2down(V2) +Jrad1down(V1) (4.3) Note that radiative coupling results in the current for sub-cell 2 containing a term dependent on sub-cell 1’s voltage, Jraddown1(V1). To identify the values of V1 and V2
that correspond to J1 and J2 being equal, subtract Jraddown1(V1) from both sides of
equation 4.3to get equation 4.4. This equation now gives a new pair of matched J-V relationships that can be solved numerically for V1 and V2 over a range of current values.
Jabs1 −Jrad1up(V1)−2Jrad1down(V1) =Jabs2 −Jrad2up(V2)−Jrad2down(V2) (4.4) Finally, Jraddown1(V1) is added back to both new J-V relationships. Performing this current-matching operation for each pair of sub-cells sequentially from highest to lowest gives a current matching J-V relation for each sub-cell, and these collectively give the J-V relationship for the entire ensemble.
Accounting for radiative coupling in series-connected ensembles can drastically change the projected performance of some band gap combinations. Figure 4.5shows the simulated J-V behavior of three different triple junction series-connected ensem- bles. The first design has band gaps that are current-matched under AM1.5D. The second is the lattice-matched Ge/GaAs/InGaP design, and the third design has band gaps chosen to be deliberately not current matched under AM1.5D. The ensemble band gaps and efficiencies without radiative coupling are shown in table 4.2.1. In all cases the red, blue and black lines correspond to the individual sub-cell J-V curves with (solid) and without (dotted) radiative coupling. The black dashed curves corre- spond to the ensemble J-V relation with radiative coupling.
The effect of radiative coupling on these designs is radically different. The current- matched ensemble has an efficiency of 47.32% without and 49.22% with radiative coupling (one sun). The lattice-matched design efficiencies are 42.45% and 44.49%, respectively. This design gets much less benefit from radiative coupling, because its total current is limited by its top and middle sub-cells. In both cases the efficiency increase comes from a slight increase in the middle and bottom sub-cell operating point. By contrast, the third design has an efficiency of 37.74% without and 46.72%
with radiative coupling. The design is severely current-limited by its middle sub-cell without radiative coupling but nicely current matched with it, and consequently it has
0 0.005 0.01 0.015 0.02
0 1 2 3 4
0 0.005 0.01 0.015
Cell voltage
Current density (A/cm2)
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Figure 4.5. Series-connected performance of candidate ideal triple junction ensembles with and without radiative coupling between sub- cells. Panel (2) shows the ensemble with optimal current matching under AM1.5D. Panel (b) shows the commercially available lattice- matched Ge/GaAs/InGaP design. Panel (c) shows a 1.03/1.42/1.8 eV design.
Top Eg Middle Eg Bottom Eg Efficiency with no coupling Efficiency with coupling
1.84 1.33 0.93 47.32% 49.22%
1.9 1.42 0.67 42.45% 44.49%
1.8 1.42 1.03 37.74% 46.72%
Table 4.1. Band gap values and efficiency for three series-connected triple junction shown in figure 4.5 both with and without radiative coupling between sub-cells.
a higher efficiency than the lattice-matched design once radiative coupling is taken into account. While the current-matched design still outperforms design 3, a designer does not always have complete freedom in choosing the band gaps of monolithically integrated sub-cells. The high efficiency of design 3’s very counter-intuitive band gap combination indicates that some designs that were previously dismissed as inefficient may deserve new consideration.
4.2.2 Optimizing series-connected ensembles for radiative cou- pling
Optimizing series-connected spectrum splitting ensembles to account for radiative coupling presents many of the same challenges as optimizing the electrically indepen- dent ensembles in Chapter 2. While the ensembles will ideally be current-matched with the radiative coupling included, there is no simple way to predict which band gap combinations will meet this requirement. Fortunately, the simulated annealing approach described in Chapter 2 is easily applied to the task. Rather than start with a randomly-generated initial design, the optimization was seeded with the op- timized series-connected ensemble from Chapter 2. The optimization again included two rounds, first with a widely-varying fluctuation applied to the ensemble and then with a narrow fluctuation. Each candidate design was evaluated for series-connected performance with radiative coupling, where 93% of each sub-cell’s total emission was assumed to be absorbed in the subsequent sub-cell. The radiative emission was cal- culated based on an air interface at the top of each sub-cell and an index-matched
2 3 4 5 6 7 8 0.5
0.55 0.6 0.65 0.7
Number of sub−cells
Efficiency
Figure 4.6. Efficiency of ideal series-connected ensembles with ra- diative coupling. Points in black correspond to ensembles optimized to be current-matched under AM1.5D, with radiative coupling ne- glected. Points in red correspond to ensembles optimized for maxi- mum efficiency with radiative coupling.
interface at the bottom, with all cells having an index of 3.6.
Figure 4.6 shows the efficiency of series-connected ensembles with radiative cou- pling with 2 to 8 sub-cell. The black curve shows the efficiency of designs optimized such that each sub-cell absorbs an equal number of photons under AM1.5D. These are the same ensembles discussed in Chapter 2. The red curve shows the efficiency of ensembles with band gaps optimized to take advantage of radiative coupling. The plot shows a significant performance advantage for designs optimized with radiative coupling in mind. The benefit increases with increasing sub-cell number, which is consistent with the trend shown in Figure 4.3 of increased benefit as sub-cell spacing decreases. The lower level of improvement for the 7 sub-cell ensemble likely comes from the increasing sensitivity of these ensembles to current starvation when one sub- cell is under-illuminated. The optimization was not able to select repeated optimum values for these ensembles, and so a higher efficiency may be attainable. However, the performance of these designs is very sensitive to slight variations in band gap value, and consequently the optimum design may require an impractical level of control over sub-cell band gap (in the proxy of material composition) for experimental realization.